11. A signal containing only two frequency components (3 kHz and 6 kHz) is sampled at the rate of 8 kHz, and then passed through a low pass filter with a cut-off frequency of 8 kHz. The filter output
12. If the signal $$x\left( t \right) = {{\sin \left( t \right)} \over {\pi t}} * {{\sin \left( t \right)} \over {\pi t}}$$ with $$ * $$ denoting the convolution operation, then x(t) is equal to
13. A sequence x(n) with the z-transform X(z) = z4 + z2 - 2z + 2 - 3z-4 is applied as an input to a linear, time-invariant system with the impulse response h(n) = 2δ(n - 3) where
$$\delta \left( n \right) = \left\{ {\matrix{
{1,} & {n = 0} \cr
{0,} & {{\rm{otherwise}}} \cr
} } \right.$$
The output at n = 4 is
$$\delta \left( n \right) = \left\{ {\matrix{ {1,} & {n = 0} \cr {0,} & {{\rm{otherwise}}} \cr } } \right.$$
The output at n = 4 is
14. The Laplace transform of i(t) is given by $$I\left( s \right) = {2 \over {s\left( {1 + s} \right)}}$$
As t → ∞, the value of i(t) tends to
As t → ∞, the value of i(t) tends to
15. A real-valued signal x(t) limited to the frequency band $$\left| f \right| \le {W \over 2}$$ is passed through a linear time invariant system whose frequency response is $$H\left( f \right) = \left\{ {\matrix{
{{e^{ - j4\pi f,}}} & {\left| f \right| \le {W \over 2}} \cr
{0,} & {\left| f \right| > {W \over 2}} \cr
} } \right.$$
The output of the system is
The output of the system is
16. The signal x(t) = sin(14000πt), where t is in seconds is sampled at a rate of 9000 samples per second. The sampled signal is the input to an ideal lowpass filter with frequency response H(t) as follows:
\[H\left( f \right) = \left\{ {\begin{array}{*{20}{c}}
{1,}&{\left| f \right| \leqslant 12kHz} \\
{0,}&{\left| f \right| > 12kHz}
\end{array}} \right.\]
What is the number of sinusoids in the output and their frequencies in kHz?
\[H\left( f \right) = \left\{ {\begin{array}{*{20}{c}} {1,}&{\left| f \right| \leqslant 12kHz} \\ {0,}&{\left| f \right| > 12kHz} \end{array}} \right.\]
What is the number of sinusoids in the output and their frequencies in kHz?
17. A signal $$2\cos \left( {{{2\pi } \over 3}t} \right) - \cos \left( {\pi t} \right)$$ is the input to an LTI system with the transfer function $$H\left( s \right) = {e^s} + {e^{ - s}}$$
If Ck denote the kth coefficient in the exponential Fourier series of the output signal, then C3 is equal to
If Ck denote the kth coefficient in the exponential Fourier series of the output signal, then C3 is equal to
18. Let $$g\left( t \right) = {e^{ - \pi {t^2}}}$$ and h(t) is a filter matched to g(t). If g(t) is applied as input to h(t), then the Fourier transform of the output is
19. A band-limited signal with a maximum frequency of 5 kHz is to be sampled. According to the sampling theorem, the sampling frequency which is not valid is
20. The result of the convolution
x(- t) ∗ δ(- t - t0) is
x(- t) ∗ δ(- t - t0) is
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